Book Review: ‘Energy Derivatives: Pricing and
Risk Management’ by Clewlow and
Strickland, 2000
Chapter 3: Volatility Estimation in Energy
Markets
Anatoliy Swishchuk
Math & Comp Lab
Dept of Math & Stat, U of C
‘Lunch at the Lab’ Talk
November 28th, 2006
Chapter 3
Chapter 3 (cntd)
Outline
 Intro
 Estimating
Volatility
 Stochastic Volatility Models
Intro


Volatility can be defined and estimated in the
context of a specific stochastic process for
the price returns
Volatility definition and measure should
capture the key features of energy markets,
such as the seasonal dependence
Intro II (most important issues)




Investment Assets vs. Consumption Goods
(Commodities cannot be treated as purely
financial assets)
Prices of Energy Commodities Display
Seasonality
Commodity Prices Often Display Jump
Behaviour
Prices Gravitate to the Cost of Production
Estimating Volatility (EV)






EV From Historical Data
EV For a Mean-Reverting Process
EV: Special Issues
Intraday Price Variability
EV for a Basket
Implied Volatility
EV from Historical Data



Step 1: Calculate Logarithmic Price Returns
Step 2: Calculate Standard Deviations of the
Logarithmic Price Returns
Step 3: Annualize the St. Dev. By Multiplying
it by the Correct Factor
EV from Historical Data II
Step 1: log price returns
(lpr)-log(1+r)
 Step 2: st. dev. of lpr
 Step 3: annualization
\sigma=sqrt(n)\sigma(lpr)
Standard usage

Seasonality effect
EV for a Mean-Reverting Process
Ornstein-Uhlenbeck process
(OU)
 Solution
 Discrete analogue
(autoregressive process)
 OU is the limiting case for
(dt->0):
\nu_t-zero mean and variance:

EV for a Mean-Reverting Process II

Recovering of the initial
parameters from
discrete version:
EV: Special Issues



The choice of the
annualisation factor and
use of intra-period data
(intraday prices)
Posibilities:
sqrt(266)=52x(4+1.107)
Sqrt(273)=52x(4+1.25)
EV: Intraday Price Variability
EV: Basket Options (Sum of 2 (weighted) or
more prices)

The Call Option Payoff:

The Put Option Payoff:
EV: Basket Options (Sum of 2 weighted
or more prices) II

Two Commodities
(GBM):

PDE:

Volatility:
Implied Volatility (IV)


IV: Vol. that is used as an input to an option pricing
formula that equates the model price with the
market price
Existence of fat tails (leptokurtic): it’s described by
the kurtosis (4th moment around the mean) (for
normal 3)
Stochastic Volatility Models (SVM)


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Ornstein-Uhlenbeck
Vasicek
Ho & Lee
Hull-White
Cox-Ingersoll-Ross
Heath-Jarrow-Morton
Continuous-time above
Stochastic Volatility Models (SVM) II
Engle (1982): ARCH(q)
Price returns
Variance


Bollerslev (1986):
GARCH(p,q)

GARCH(1,1):
Stochastic Volatility Models (SVM) IV
Stochastic Volatility Models (SVM) III
EV: Estimation and Testing
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
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Parameters
Estimation
Usefulness of a
parameter estimator:
Unbiased and Efficient
Unbiased is good
Biased but Efficient
may be preferable to
an unbiased
Estimation and Testing: Least Squares

Stochastic equation:

Minimization:
Estimation and Testing: Least Squares II

Example I:

Estimation of Mean
Estimation and Testing: Least Squares II

Example II:

Estimation of
Standard
Deviation
Unbiased,
consistent,
efficient

Maximum Likelihood Estimation (MLE)

Equation:

Probability density
function:

Joint distribution:

Likelihood
function:
MLE I

Maximising
Equations
are:
MLE II

MLE for St. Dev.:


Consistent
But biased

Unbiased (LSE)
Testing
Testing II

Skewness

Kurtosis

Jarque-Bera Statistic

Goldfeld-Quandt test
Testing (Example from Energy
Commodity Markets)
Testing (Example from Energy
Commodity Markets I)
Testing (Goodness of Fit)

Likelihood Ratio Test:

Schwartz Criterion (SC)
(the most probable
model-with the smallest
SC):

Testing (Goodness of Fit)
Testing (Goodness of Fit)
Figures (Simulated vs. Actual Data): PD
Figures (Simulated vs. Actual Data): JD
Figures (Simulated vs. Actual Data):
JD+GARCH
The End

Thank You